Chapter 9

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# Chapter 9 - PowerPoint PPT Presentation

Chapter 9. Linear Momentum and Collisions. Intro. Consider bowling: Bowling ball collides with initial pin Force on/Acceleration of the Pin Force on/Acceleration of the ball Momentum- simplified way to study these moving objects. 9.1 Linear Momentum and Conservation.

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### Chapter 9

Linear Momentum and Collisions

Intro
• Consider bowling:
• Bowling ball collides with initial pin
• Force on/Acceleration of the Pin
• Force on/Acceleration of the ball
• Momentum- simplified way to study these moving objects.
9.1 Linear Momentum and Conservation

Consider two particles (isolated) m1 and m2 moving at v1 and v2

From Newton’s 3rd Law

9.1
• And if the the masses are constant
• If the derivative of a function is 0 it is constant (conserved)
9.1
• Linear Momentum- the product of the mass and velocity of a moving particle.
• Momentum is a vector quantity
• Has dimensions MLT-1 and SI units kg.m/s
• All momentum is conserved
• Three Components of Momentum
9.1
• Directly related to Newton’s 2nd Law
• Instead of Net Force equals mass times accel.
• Can be described as Net Force equals time rate of change of momentum
9.1
• Conservation of Momentum
• For isolated systems the time derivative of the total momentum is 0
• The total momentum is therefore constant or conserved.

Law of Conservation of Momentum

• And in 3 components
9.1
• Quick Quizzes p 254-255
• Examples 9.1-9.2
9.2 Impulse and Momentum
• The momentum of an object changes when a net force acts on it

or

• Integrating this gives
9.2
• Impulse-Momentum Theorem- the impulse of the force F acting on a particle equals the change in momentum of the particle
• Force Varies, impulse time is short impulse can generally be calculated with the average force.
9.2
• Quick Quizzes p. 258
• Examples 9.3-9.4
9.3 Collisions in 1-D
• Momentum is conserved
• Three types of Collisions
• Inelastic
• Perfectly Inelastic
• Elastic
9.3
• Inelastic Collision- Momentum is conserved but kinetic energy is not.
• Inelastic- objects collide and separate, some K is lost
• Perfectly Inelastic- objects collide and stick together (moving as one), some K is lost
9.3
• Elastic Collisions-A collision in which no energy is lost to (surroundings / internal / potential)
• Both momentum and kinetic energy are conserved
9.3
• By combining the Conservation of p and K equations
• In all collision types careful attention to the direction (and sign) of velocities must be paid.
9.3
• Quick Quizzes p. 262
• Examples 9.5 - 9.9
9.4 2-D Collisions
• Momentum is conserved on each axis
• Examples 9.10 – 9.12
9.5 Center of Mass
• We can describe the overall motion of a mechanical system by tracking its center of mass
• System could be a group of particles
• System could be a large extended object
• A force applied to the center of mass will cause no rotation to the system
9.5
• To find the center of mass in 3-D space for a number (i) particles
• Or in terms of the position vector of each particle
9.5
• For extended objects that have a continuous mass distribution
• Consider them an infinite number of closely spaced particles
• The sum becomes an integral
9.5
• Or in terms of the position vector
• For symmetrical objects, the center of mass lies on the axis/plane of symmetry
• Examples: uniform rod,

sphere,

cube,

donut?

9.5
• For extended objects, the force of gravity acts individually on each small piece of mass (dm)
• The net effect of all these forces is equivalent to the single force Mg, through a point called the center of gravity.
• If the gravitational field is uniform across all dm, the center of gravity and center of mass are one and the same.
9.5
• Quick quiz p 272
• Examples 9.13, 9.14, 9.15
9.6 Motion of a System
• If the mass of a system remains constant (no particles entering/leaving) then we can track the motion of the center of mass, rather than the individual particles.
• Also assumes any forces on the system are internal (isolated)
9.6
• Velocity of the center of mass
• Acceleration of the center of mass
9.6
• If there is a net force on the system, it will move equivalent to the way a single M with the same net force would move.
• And if the net force is zero
9.6
• Quick Quizzes p. 276
• Examples 9.17, 9.18
9.7 Rocket Propulsion
• Most forms of vehicular motion result from action/reaction friction.
• A rocket has nothing to push against so its motion/control depend on conservation of motion of the system.
• The system includes the rocket body (and payload) plus the ejected fuel
9.7
• The rocket burns fuel and oxidizer creating expanding gases that are directed through the nozzle.
• Each gas molecule has a mass (that was once part of the rockets total mass) and velocity, therefore a downward momentum.
• The rocket receives the same compensating momentum upward.
9.7
• Looking a rocket initially with mass M + Δm, moving with velocity v…
9.7
• And some time, Δt, later...
• The rocket now has mass, M and velocity v + Δv, compensating the momentum of the exhausted mass, Δm.
9.7
• The conservation of momentum expression for this change…
• Can be simplified to…
9.7
• A rocket motor produces a continuous flow of exhaust gas a fairly constant speed, through the burn
• For continually changing values…

Δv dv

Δm dm

So… 

9.7
• Because the increase in exhaust mass = the decrease in rocket mass…
• Then integrate this expression
9.7
• Discuss integral of M-1
• Evaluating from vi to vf gives the basic expression for rocket propulsion.
9.7
• Mi is the total mass of the rocket/payload plus fuel
• Mf is the mass of the rocket/payload
• Mi – Mf is the mass of fuel needed to achieve a certain speed (eg. Escape speed to power down rocket)
9.7
• Thrust- the actual force on the rocket at any given time is
• Thrust is proportional to exhaust speed and also the rate of change of mass (burn rate).
• Examples 9.19 p. 279